Forests, cumulants, martingales

Realized cumulants for martingales

Generalizing the realized variance, the realized skewness (Neuberger, 2012) and the realized kurtosis (Bae and Lee, 2020), we construct realized cumulants with the so-called aggregation property. They are unbiased statistics of the cumulants of a martingale marginal based on sub-period increments of the martingale and its lower-order conditional cumulant processes. Our key finding is a relation between the aggregation property and the complete Bell polynomials. For an application we give an alternative proof and an extension of a cumulant recursion formula recently obtained by Lacoin et al. (2…

A forest formula for pre-Lie exponentials, Magnus' operator and cumulant-cumulant relations

Forest formulas that generalize Zimmermann's forest formula in quantum field theory have been obtained for the computation of the antipode in the dual of enveloping algebras of pre-Lie algebras. In this work, largely motivated by Murua's analysis of the Baker-Campbell-Hausdorff formula, we show that the same ideas and techniques generalize and provide effective tools to handle computations in these algebras, which are of utmost importance in numerical analysis and related areas. We illustrate our results by studying the action of the pre-Lie exponential and the Magnus operator in the free pre-…

A Cluster Expansion Proof That The Stochastic Exponential Of A Brownian Motion Is A Martingale

Let $ψ:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a smooth and continuous real function and $ψ\in\mathrm{L}^{2}(\mathbb{R}^{+})$. Let ${B}(t)$ be a standard Brownian motion defined with respect to a probability space $(Ω,\mathscr{F},{\mathsf{P}})$ and where $d{B}(t)=ξ(t)dt$ and $t\in\mathbb{R}^{+}$. The process $ξ(t)$ is a Gaussian white noise with expectation $\mathbf{\mathsf{E}}~ξ(t)=0$ and with covariance ${\mathsf{E}}~ξ(t)ξ(s)=δ(t-s)$. The Dolean-Dades stochastic exponential ${Z}(t)$ is the solution to the linear stochastic differential equation describing a geometric Brownian motion such…

与Lévy过程相关的多重随机积分的乘积公式

Product formulas for multiple stochastic integrals associated with Lévy processes

本文针对与Lévy过程相关的随机测度多重积分，利用补偿协变稳定族鞅的表示公式建立显式乘积公式，给出核函数的充分条件，并应用于期望、矩、累积量计算及中心极限定理。